Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Use a formula to find the sum of each arithmetic series. $$1+3+5+7+\cdots+97$$

Step-by-Step Solution

Answer

The sum of the arithmetic series is 2401.

Step 1: Identify the sequence characteristics

First, we identify the first term (\(a\)) and the common difference (\(d\)) in the given arithmetic series. The series is given as \(1 + 3 + 5 + 7 + \, \cdots \, + 97\). Here, the first term \(a = 1\), and the common difference \(d = 3 - 1 = 2\).

Step 2: Find the number of terms

Next, we need to find the total number of terms (\(n\)) in the series. We use the formula for the last term of an arithmetic sequence: \(l = a + (n-1)\cdot d\). Plugging in the values, we have \(97 = 1 + (n-1)\cdot 2\). Simplifying gives \(97 - 1 = 2(n-1)\), which simplifies further to \(96 = 2(n-1)\) and \(n = 49\).

Step 3: Use the sum formula for an arithmetic series

The formula to find the sum of \(n\) terms of an arithmetic series is \(S_n = \frac{n}{2} \cdot (a + l)\), where \(a\) is the first term and \(l\) is the last term. Substituting the known values: \(S_{49} = \frac{49}{2} \cdot (1 + 97)\).

Step 4: Calculate the sum

Calculate \(S_{49}\) as follows: \(S_{49} = \frac{49}{2} \cdot 98\). Simplifying, we have \(S_{49} = 49 \cdot 49\), which results in a sum of \(S_{49} = 2401\).

Key Concepts

Sum of Series Formula

The sum of an arithmetic series can be calculated using a straightforward formula. This formula helps us find the sum of a series by relating the number of terms, the first term, and the last term. The formula is:

• \( S_n = \frac{n}{2} \cdot (a + l) \)

Here, \( S_n \) represents the sum of the series. The variable \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. By using this formula, one can easily calculate the sum without manually adding each term.
For example, in a series where the first term is 1, the last term is 97, and the number of terms is 49, you can quickly find the sum by substituting these values into the formula. This method streamlines the calculation process.

Common Difference

In an arithmetic series, the common difference is a critical component. It refers to the constant difference between each pair of consecutive terms in the series. To find the common difference \( d \) in a sequence:

• Subtract the first term from the second term: \( d = a_2 - a_1 \)

Using the example series, starting with 1 and proceeding with 3, 5, 7, etc., the common difference is \( d = 3 - 1 = 2 \).
This constant difference creates the backbone of the arithmetic sequence, ensuring each term follows a predictable pattern.
Identifying the common difference is essential, as it allows us to ascertain other sequence properties, like predicting future terms or calculating the sum.

Number of Terms

Determining the number of terms in an arithmetic sequence is crucial for properly using the sum formula. To find the number of terms \( n \), you can use the last term formula:

• \( l = a + (n-1) \cdot d \)

Rearrange the formula to solve for \( n \), given \( l \) (the last term), \( a \) (the first term), and \( d \) (the common difference).
In the example series ending in 97, with a start at 1 and a common difference of 2, solve the equation \( 97 = 1 + (n-1) \cdot 2 \) to find \( n \). Simplify to get \( n = 49 \).
Knowing \( n \) is vital because it allows us to use the sum formula effectively to calculate the total sum of the series.

Sequence Characteristics

Understanding the characteristics of an arithmetic sequence helps in analyzing and computing its various properties. Characteristics include:

• The first term \( a \), which is the starting point of the series.
• The common difference \( d \), denoting the uniform spacing between terms.
• The formula for any specific term in the sequence, \( a_n = a + (n-1) \cdot d \).

These characteristics allow us to frame the sequence, predict future terms, or find particular terms' values.
For instance, to find the 30th term in a series starting from 1, with a common difference of 2, use the formula: \( a_{30} = 1 + 29 \cdot 2 = 59 \).
These insights are not just academic; they provide practical applications in situations like financial planning or architectural design, where predictable patterns matter.